We review some typical problems in Hydrogeology, Plasticity, and Biomechanics while looking for connections with the tool of Fractional Calculus [1].We shall consider some fundamental issues of the mentioned research fields, and mention the challenges launched by the necessity of dedicated experiments, reliable mathematical modelling, and high-performance computing. Hydrogeology: A leading topic in hydrogeology is the investigation of groundwater flow. Some typical problems are sea-water intrusion into coastal aquifers, upconing of saline water from deep aquifers, and flow around salt domes. When these phenomena occur, the density of the fluid is generally not uniform. Rather, it is a function of pressure, temperature, and pressure. The deviation of the fluid mass density from a reference uniform value generates a flow which is said to be density-driven. The equations of density-driven flow are non-linear and coupled. This often leads to non-uniqueness of solutions and stability problems (2). Plasticity: We consider a biaxial Hopkinson's bar in the case in which this device consists of a cruciform tensile specimen surrounded by four long elastic bars. We study plastic waves inside the specimen. Our approach is based on the quasi rate-independent model put forward in (3). We show some preliminary results for initial and subsequent elastic ranges under both symmetric and non-symmetric boundary conditions. Biomechanics: We discuss some aspects of growth, mass transfer, and remodelling of biological tissues that can be macroscopically described by multiphasic media. Our purposes are to define evolution laws for the growth and the remodelling variables, and to characterize the thermodynamic equilibrium of the system. Our approach is based on the papers (4)(5). [1] Atanackovic, T.M. (2002). Continuum Mech. Thermodyn., 14, 137{148. [2] Diersch, H.-J- G., Kolditz O. (2005). Wasy Software FEFLOW c - Finite Element Subsurface Flow and Transport Simulation System, vol. II. Wasy GmbH. [3] Micunovic, M.V. (2009). Thermodynamics of Viscoplasticity: Fundamentals and Applications. Springer-Verlag, Heidelberg, New York. [4] DiCarlo, A., Quiligotti, S. (2002). Mech. Res. Comm., 29, 449-456. [5] Grillo, A., et al. (2009). Nuovo Cimento C, 32(1), 97-119.